Minimal rings related to generalized quaternion rings

Year 2023, , 88 - 111, 10.07.2023

Jose Maria Grau Antonio M. Oller-marcen Steve Szabo

https://doi.org/10.24330/ieja.1281705

Abstract

The family of rings of the form
\frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left \langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \right \rangle}
is investigated which contains the generalized Hamilton quaternions over $\Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $\Z_{p^s}$ are characterized.

Keywords

Reversible ring, symmetric ring, reflexive ring, abelian ring, semicommutative ring, generalized quaternion ring

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